\magnification = 2000 
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros


\def\ni{\noindent}

\vglue -2pt


\Title Tractrix.
     
\ni
The Tractrix is a curve with the following nice interpretation:
Suppose a dog-owner takes his pet along as he goes for  a walk
``down'' the $y$-axis.  He starts from the origin, with his dog initially 
standing on the $x$-axis at a distance aa away from the owner. 
Then the Tractrix is the path followed by the dog if he ``follows his 
owner unwillingly'', i.e.,  if he constantly pulls against the leash, 
keeping it tight. This means mathematically that the leash is always 
tangent to the path of the dog, so that the length of the tangent 
segment from the Tractrix to the y-axis has constant length aa. 
Parametric equations for the Tractrix (take $bb=0$) are: 
 

$x(t) = aa \cdot \sin(t)(1+bb)$
 
$y(t) = aa \cdot (\cos(t)(1+bb) + \ln(\tan(t/2)) )$.
\LF
The curves obtained for $bb \ne 0$ are generated by the same kinematic
motion, except that a different point of the moving plane is taken as the
drawing pen. See the default {\tt Morph}.\lf
The Tractrix has a well-known surface of revolution, called the
Pseudosphere, Namely, rotating it around the y-axis gives a surface with
Gaussian curvature -1. This means that the Pseudosphere can
be considered as a portion of the Hyperbolic Plane. The latter 
is a geometry that was discovered in the 19th century by Bolyai 
and Lobachevsky. It satisfies all the axioms of Euclidean Geometry 
except the Axiom of Parallels. In fact, through a point outside a
given line (= geodesic) there are infinitely many lines that are 
parallel to (i.e., do not meet) the given line.

\ni
There are many connections, sometimes unexpected, between planar
curves. For the Tractrix select: {\tt Show Osculating Circles And Normals}.
One observes a Catenary (see another entry in the curve menu) as the 
envelope of the normals.

\ni
H.K.


\bye